The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs

Abstract

Let q(x, t) satisfy a nonlinear integrable evolution PDE whose highest spatial derivative is of order n. An initial boundary value problem on the half-line for such a PDE is at least linearly well-posed if one prescribes initial conditions, as well as N boundary conditions at x = 0, where for n even N equals n 2 and for n odd, depending on the sign of the highest derivative, N equals either n−1 2 or n+1 2 . For example, for the nonlinear Schrödinger (NLS) and the sine-Gordon (sG), N = 1, while for the modified Korteweg-deVries (mKdV) N = 1 or N = 2 depending on the sign of the third derivative. Constructing the generalized Dirichlet-to-Neumann map means determining those boundary values at x = 0 that are not prescribed as boundary conditions in terms of the given initial and boundary conditions. A general methodology is presented that constructs this map in terms of the solution of a system of two nonlinear ODEs. This formulation implies that for the focusing NLS, for the sG, and for the two focusing versions of the mKdV, this map is global in time. It appears that this is the first time in the literature that such a characterization for nonlinear PDEs is explicitly described. It is also shown here that for particular choices of the boundary conditions the above map can be linearized.

Integrable Nonlinear Evolution Equations on a Finite Interval

“…This yields q x (0, t) in terms of a system of four nonlinear ODEs satisfied by the functions {L j ,M j } 2 1 . A further important development is presented in [9], where it is shown that it is possible to express {L j ,M j } 2 1 in terms of Φ 1 and Φ 2 . Thus the formalism presented in [9] expresses q x (0, t) in terms of a system of two nonlinear ODEs satisfied by the functions Φ 1 and Φ 2 .…”

“…A further important development is presented in [9], where it is shown that it is possible to express {L j ,M j } 2 1 in terms of Φ 1 and Φ 2 . Thus the formalism presented in [9] expresses q x (0, t) in terms of a system of two nonlinear ODEs satisfied by the functions Φ 1 and Φ 2 . Similarly, it is shown in [9] that the unknown boundary values for the sG, the mKdV I, and the mKdV II equations can also be expressed in terms of a system of two nonlinear ODEs.…”

“…Thus the formalism presented in [9] expresses q x (0, t) in terms of a system of two nonlinear ODEs satisfied by the functions Φ 1 and Φ 2 . Similarly, it is shown in [9] that the unknown boundary values for the sG, the mKdV I, and the mKdV II equations can also be expressed in terms of a system of two nonlinear ODEs. Furthermore, it is shown in [9] that this system for the sG and for the focusing versions of the NLS,the mKdV I, and the mKdV II equations has a global solution.…”

“…In the present paper, using the general methodology of [6] and the recent results of [9], we analyze the following IBV problems:…”

“…The analysis of Step 3 is based on the Gelfand-Levitan-Marchenko representation of the eigenfunctions Φ(t, k) and Ψ(t, k). For example, in the case of the sG equation, it can be shown [9] that Φ can be expressed in terms of four functions {M j (t, s), L j (t, s)} 2 1 , −t < s < t, t > 0, satisfying a hyperbolic system of four PDEs as well as the Goursat boundary conditions on the characteristics:…”